Generalized Memory Polynomial (GMP)

The GMP serves as a pruned Volterra series, retaining only the most physically significant kernel slices while discarding redundant off-diagonal terms.

• Achieves modeling accuracy comparable to full Volterra with dramatically fewer coefficients
• Typical GMP configuration uses 3-5 memory taps and 5-7 nonlinearity orders
• Coefficient count scales as $O(M \cdot K \cdot L)$ rather than $O(M^K)$ for full Volterra
• Enables real-time FPGA implementation with manageable resource utilization

Each GMP basis function is constructed from a signal term multiplied by an envelope power term at different time offsets, creating a rich set of regressors.

• Type 1 (aligned): $x(n-q) \cdot |x(n-q)|^k$ — classical memory polynomial terms
• Type 2 (lagging): $x(n-q) \cdot |x(n-q-m)|^k$ — envelope memory lagging behind signal
• Type 3 (leading): $x(n-q) \cdot |x(n-q+m)|^k$ — envelope memory leading the signal
• This taxonomy enables systematic model selection based on device physics

For wideband signals exceeding 100 MHz instantaneous bandwidth, the GMP significantly outperforms the standard memory polynomial in capturing frequency-dependent memory effects.

• Reduces NMSE by 3-5 dB compared to memory polynomial for GaN Doherty PAs with 200 MHz LTE signals
• Accurately models long-term thermal memory through extended cross-term delays
• Captures bias modulation effects that manifest as envelope-frequency interactions
• Essential for 5G NR signals with 400 MHz+ carrier bandwidths

The GMP basis matrix can exhibit high condition numbers due to correlation between cross-terms and standard memory polynomial terms, requiring careful regularization.

• Apply ridge regression (L2 regularization) to stabilize coefficient extraction
• Use orthogonalization techniques like modified Gram-Schmidt to decorrelate basis functions
• Implement variable selection algorithms to prune redundant cross-terms
• Monitor condition number during online adaptation to detect numerical instability

The GMP framework naturally extends to concurrent multi-band scenarios by including cross-modulation products between carrier frequencies.

• 2D-GMP processes dual-band signals with inter-band and cross-band modulation terms
• Captures intermodulation distortion falling within and between transmit bands
• Supports non-contiguous carrier aggregation configurations in 5G NR
• Enables single predistorter linearization of multi-band Doherty transmitters

The mathematical foundation upon which the GMP is built. It models nonlinear dynamic systems using a series of multidimensional convolution integrals. While theoretically complete, its exponential complexity growth with memory depth and nonlinearity order makes it impractical for real-time DPD. The GMP is a pruned, computationally efficient subset of the full Volterra series, retaining only the most significant cross-terms.

A simplified Volterra subset that models nonlinearity with memory using diagonal terms only. It captures the interaction between a signal sample and its own envelope powers at different delays. While computationally efficient, the MP often fails to capture complex cross-memory effects in wideband or Doherty amplifiers. The GMP extends the MP by adding lagging/leading cross-terms between delayed samples and the envelope of other samples.

An alternative Volterra simplification that separates static nonlinearity from low-order dynamic deviation terms. Unlike the GMP, which prunes based on term structure, DDR reduces complexity by limiting the dynamic order. It is particularly effective for amplifiers where memory effects are small perturbations around a static nonlinearity. Both DDR and GMP aim to balance model fidelity against coefficient count.

A slowly varying change in amplifier gain and phase caused by self-heating dependent on signal history. These long-term memory effects manifest as asymmetric intermodulation distortion products. The GMP's inclusion of cross-terms between different time lags helps model these complex, non-quasi-static behaviors that simpler memory polynomials cannot capture, especially in GaN-based high-power amplifiers.

Slow charge capture and release phenomena in semiconductor materials like GaN. These cause long-term memory effects and dynamic nonlinear behavior that depend on the peak-to-average ratio and envelope history of the signal. GMP models with sufficient memory depth and cross-term order are often required to accurately linearize amplifiers exhibiting significant trapping-induced distortion.

The GMP is a linear-in-parameters model, meaning its coefficients can be extracted using standard least-squares techniques:

• Least Squares (LS): Direct matrix inversion for offline extraction
• Recursive Least Squares (RLS): Adaptive algorithm for online coefficient updates
• Regularized LS: Ridge regression or LASSO to ensure numerical stability when the GMP basis matrix becomes ill-conditioned due to correlated terms